Question

Polymerization equilibrium. Consider the process of polymerization and assume the principle of equal reactivity, that each monomer adds with the same equilibrium constant as the previous one: $$\begin{array}{c}2X_1\stackrel{K}{⟶}{X}_2,\\ 3X_1\stackrel{K^2}{⟶}{X}_3,\\ ⋮\\ n{X}_1\stackrel{K^{m-1}}{⟶}{X}_n.\end{array}$$ (a) Write the binding polynomial $Q$. (b) Write the average chain length $⟨n⟩$ in terms of $Q$. (c) Plot $⟨n⟩$ versus $x$ for $K=1$ and for $Kx<1$.

Short answer

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Step-by-step solution

- Write the binding polynomial Q

The binding polynomial (also known as the partition function) is given by the sum of all possible states that the system can adopt. For the polymerization process, each state corresponds to the formation of a polymer chain of length n with an equilibrium constant that depends on n. Given the equilibrium constants, the binding polynomial can be written as: $$Q=\text{sum of all states}=1+\frac{K[X_1{]}^2}{2!}+\frac{K^2[X_1{]}^3}{3!}+\frac{K^3[X_1{]}^4}{4!}+\frac{K^4[X_1{]}^5}{5!}+\to \text{ up to }n\text{ terms}$$

- Write the average chain length ⟨n⟩ in terms of Q

The average chain length $⟨n⟩$ is the average number of monomers in the polymer chain. It can be expressed in terms of the binding polynomial Q as: $$⟨n⟩=\frac{\text{total sum of chain lengths}}{Q}=\frac{\sum _{n=1}^{\text{and term}}n\times Q_n}{Q}$$

Key concepts

binding polynomial

In polymerization equilibrium, the binding polynomial, also known as the partition function, is an essential concept. It represents the sum of all possible states the polymer system can adopt.
For polymer chains, each state corresponds to a polymer of a certain length forming with a specific equilibrium constant. Given the equilibrium constants, the binding polynomial can be formulated as follows:

The basic form of the binding polynomial is:
$$Q=1+\frac{K[X_1{]}^2}{2!}+\frac{K^2[X_1{]}^3}{3!}+\frac{K^3[X_1{]}^4}{4!}+\frac{K^4[X_1{]}^5}{5!}+\to \text{ up to }n\text{ terms}$$

Here, the first term $1$ represents the monomer itself, and each successive term represents polymer chains of length increasing by one monomer at a time.

This polynomial helps us understand how the polymers of various lengths contribute to the overall equilibrium.
Remember that in these expressions, $K$ is the equilibrium constant for the polymerization step, and $[X_1]$ is the concentration of the monomer.
This comprehensive sum describes all possible configurations from monomers to large chains.

average chain length

The average chain length $⟨n⟩$ signifies the mean number of monomers in a polymer chain. This is crucial as it indicates the extent of polymerization in the system.
To determine this, we consider the binding polynomial $Q$ and the sum of all chain lengths weighted by their occurrences:
$$⟨n⟩=\frac{\text{total sum of chain lengths}}{Q}=\frac{\sum _{n=1}^{\text{end term}}n\times Q_n}{Q}$$
This formula breaks down as follows:

• $n$ represents the number of monomers in the chain.
• $Q_n$ is the term representing the weight of a polymer chain of length $n$.
• $Q$ is the binding polynomial, which normalizes the total sum.

For instance, if you have chains up to length $5$, the terms within the summation would be $1\times Q_1,2\times Q_2$, and so forth. This calculation ultimately gives the expected length of polymers in the system.
Thus, seeing changes in $⟨n⟩$ under different conditions can provide insights into how the polymerization process behaves.

equilibrium constants

Equilibrium constants $(K)$ play a central role in understanding polymerization equilibrium.
Each step in the polymerization process has an associated equilibrium constant, which describes the ratio of the concentrations of products to reactants at equilibrium. For polymer chains:

• $2X_1\stackrel{K}{⟶}{X}_2$
• $3X_1\stackrel{K^2}{⟶}{X}_3$
• ... and so forth.

In this scheme, the equilibrium constant for the formation of a dimer is $K$, a trimer involves $K^2$, and so on.
These constants highlight the progressive nature of polymerization, where each additional monomer joins with a dependence on the previous equilibrium constants.
Understanding these equilibrium constants is vital as they affect the formation and stability of different polymer chain lengths, directly influencing the binding polynomial $(Q)$ and the average chain length $⟨n⟩$.